Lower Multiplicity for Irreducible Representations of Nilpotent Locally Compact Groups

Kaniuth, Eberhard

doi:10.1006/jfan.1999.3469

Cited by 2 publications

(2 citation statements)

References 20 publications

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“…Consequently, we then use both sides of this link to compute the upper multiplicities of all irreducible representations of the groups G N (N 3) in the``threadlike'' generalisation of the Heisenberg group (see Section 3). Other applications of the C*-algebraic theory of [1,4,5] to irreducible representations of locally compact groups have previously been given in [2,11,17].…”

Section: Introductionmentioning

confidence: 99%

Strength of Convergence in Duals of C*-Algebras and Nilpotent Lie Groups

Archbold¹,

Kaniuth²,

Ludwig³

et al. 2001

Advances in Mathematics

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By using trace formulae, the recent concept of upper multiplicity for an irreducible representation of a C*-algebra is linked to the earlier notion of strength of convergence in the dual of a nilpotent Lie group G. In particular, it is shown that if ? # G has finite upper multiplicity then this integer is the greatest strength with which a sequence in G can converge to ?. Upper multiplicities are calculated for all irreducible representations of the groups in the threadlike generalization of the Heisenberg group. The values are computed by combining new C*-theoretic results with detailed analysis of the convergence of coadjoint orbits and they show that every positive integer occurs for this class of groups.

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Section: Introductionmentioning

confidence: 99%

Strength of Convergence in Duals of C*-Algebras and Nilpotent Lie Groups

Archbold¹,

Kaniuth²,

Ludwig³

et al. 2001

Advances in Mathematics

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“…The papers [2,5,7,8] focussed on the failure of the Hausdorff property and studied individual points in G, ranging from separated points to points in the cortex (the set of irreducible representations which cannot be separated from the trivial representation by disjoint open sets). On the other hand, the papers [6,14] focussed on the second problem and studied, for each π ∈ G, the greatest and least strength with which a sequence could converge to π. This provided a link with the notions of upper and lower multiplicity for π : M U (π) and M L (π) ( [1,3,4]).…”

Section: Introductionmentioning

confidence: 99%

Limit sets and strengths of convergence for sequences in the duals of thread-like Lie groups

Archbold

Ludwig²,

Schlichting

2006

Math. Z.

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We consider a properly converging sequence of non-characters in the dual space of a thread-like group G N (N ≥ 3) and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if (π k ) is a properly convergent sequence of non-characters in G N , then there is a trade-off between the number of limits σ which are not characters, their degrees, and the strength of convergence i σ to each of these limits (Theorem 3.2). This enables us to describe various possibilities for maximal limit sets consisting entirely of non-characters (Theorem 4.6). In Sect. 5, we show that if (π k ) is a properly converging sequence of non-characters in G N and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to R 2 ) with the limit set has at most two components. In the case of two components, each is a half-plane. In Theorem 7.7, we show that if a sequence has a character as a cluster point then, by passing to a properly convergent subsequence and then a further subsequence, it is possible to find a real null sequence (c k ) (with c k = 0) such that, for a in the Pedersen ideal of C * (G N ), lim k→∞ c k Tr(π k (a)) exists (not identically zero) and is given by a sum of integrals over R 2 .

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Lower Multiplicity for Irreducible Representations of Nilpotent Locally Compact Groups

Cited by 2 publications

References 20 publications

Strength of Convergence in Duals of C*-Algebras and Nilpotent Lie Groups

Strength of Convergence in Duals of C*-Algebras and Nilpotent Lie Groups

Limit sets and strengths of convergence for sequences in the duals of thread-like Lie groups

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